Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 1 / 10 / b / iii / Solution

Set $\psi (r)=u(r)/r$ for $r>0$. Direct differentiation gives $(r^2{\psi }^{\prime }{)}^{\prime }=r{u}^{\prime \prime }$, so the radial equation reduces toBoundedness of $\psi$ requires $u(0)=0$, and the wall condition requires $u(a)=0$. The nontrivial solutions are consequentlyTheir origin values are the removable limits $C_{j}j\pi /a$. At $k=0$, the only bounded radial solution is constant and the wall forces it to vanish, so there is no zero mode. The lowest eigenvalue isIt is below $10.5/a^2$, as required by the Rayleigh-Ritz variational principle. The trial value overestimates $k_0^2$ by approximately $\boxed{6.39\%}$.